# Aubry-Mather Theory for Lorentzian Manifolds

### (Regensburg University)

**Event:** ERC School on Analysis in Metric Spaces and Geometric Measure Theory

**Date:** Jan 13, 2011,
**time:** 18:10

**Place:** Centro De Giorgi, Scuola Normale Superiore

**Abstract.** I will introduce a new class of compact spacetimes, called class Aspacetimes and show how Aubry-Mather theory can be adapted to Lorentzian geometry. In the first part I define class A spacetimes and studythe important causality theoretic aspects. This includes providing therelevant objectsand notions for the subsequent discussion. In the second part I explain the Aubry-Mather theory for class A spacetimes. Aubry-Mather theory studiesthe asymptotic behavior of homologically extremal orbits of certainEuler-Lagrange flows. Homological extremal means that the base curvesof these orbitslift to global extremals in the Abelian cover. In Lorentzian geometryone therefore studies causal geodesics which lift to maximal ones. I give existence and multiplicity results for timelike lines in the Abelian cover.